Related papers: Convex Hull Formation for Programmable Matter
A class of algorithms for the solution of discrete material optimization problems in electromagnetic applications is discussed. The idea behind the algorithm is similar to that of the sequential programming. However, in each major iteration…
Biological and robotic systems often operate in confined environments where material must be gathered without centralized control. Inspired by the effective collection strategies of aquatic worms (Lumbriculus variegatus and Tubifex…
We study a natural extension to the well-known convex hull problem by introducing multiplicity: if we are given a set of convex polygons, and we are allowed to partition the set into multiple components and take the convex hull of each…
This paper develops an algorithm that guides a multi-robot system in an unknown environment in search of fixed targets. The area to be scanned contains an unknown number of convex obstacles of unknown size and shape. The algorithm covers…
We explore the use of templated self-assembly to facilitate the formation of complex target structures made from patchy particles. First, we consider the templating of high-symmetry shell structures around a spherical core particle. We find…
In this paper we present several results on the expected complexity of a convex hull of $n$ points chosen uniformly and independently from a convex shape. (i) We show that the expected number of vertices of the convex hull of $n$ points,…
The experiments conducted by various scientific groups indicate that, in dense two-dimensional systems of elongated particles subjected to vibration, the pattern formation is possible. Computer simulations have evidenced that the random…
Here, we present a novel algorithm that discriminates between bound and unbound particles by consideration of the gravitational potential from an accelerated reference frame -- also referred to as `the boosted potential'. Particles are…
We study the convex hull of $SO(n)$, thought of as the set of $n\times n$ orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of $SO(n)$ is doubly spectrahedral, i.e.…
We present a mathematical and algorithmic scheme for learning the principal geometric elements in an image or 3D object. We build on recent work that convexifies the basic problem of finding a combination of a small number shapes that…
There are many space subdivision and space partitioning techniques used in many algorithms to speed up computations. They mostly rely on orthogonal space subdivision, resp. using hierarchical data structures, e.g. BSP trees, quadtrees,…
Online convex optimization is a sequential prediction framework with the goal to track and adapt to the environment through evaluating proper convex loss functions. We study efficient particle filtering methods from the perspective of such…
The convex hull of a data set $P$ is the smallest convex set that contains $P$. In this work, we present a new data structure for convex hull, that allows for efficient dynamic updates. In a dynamic convex hull implementation, the following…
An incremental approach for computation of convex hull for data points in two-dimensions is presented. The algorithm is not output-sensitive and costs a time that is linear in the size of data points at input. Graham's scan is applied only…
Particle-based dynamic occupancy maps were proposed in recent years to model the obstacles in dynamic environments. Current particle-based maps describe the occupancy status in discrete grid form and suffer from the grid size problem,…
Individual modules of programmable matter participate in their system's collective behavior by expending energy to perform actions. However, not all modules may have access to the external energy source powering the system, necessitating a…
Dark matter halos are typically defined as spheres that enclose some overdensity, but these sharp, somewhat arbitrary boundaries introduce non-physical artifacts such as backsplash halos, pseudo-evolution, and an incomplete accounting of…
Recent advancements in the synthesis of anisotropic macromolecules and nanoparticles have spurred an immense interest in theoretical and computational studies of self-assembly. The cornerstone of such studies is the role of shape in…
In this paper, we first consider the subpath convex hull query problem: Given a simple path $\pi$ of $n$ vertices, preprocess it so that the convex hull of any query subpath of $\pi$ can be quickly obtained. Previously, Guibas, Hershberger,…
We consider offsets of a union of convex objects. We aim for a filtration, a sequence of nested cell complexes, that captures the topological evolution of the offsets for increasing radii. We describe methods to compute a filtration based…